While our results indicate that money growth is strongly correlated with current inflation, we have not directly tested the hypothesis that low-frequency move-ments in money growth cause inflation. This is important because the finding that money growth and inflation move together may simply reflect a stable money-demand function and need not imply that money growth causes inflation.
To understand properly the inflation process an understanding of the patterns of causality is consequently needed.
30 In order not to be overly restrictive we do not impose the coefficients obtained in the pre-vious section on equation (9), but reestimate all coefficients and test for the long-run validity of the quantity theory.
31 Woodford (2007) shows that a New-Keynesian Phillips curve model together with a stan-dard money demand equation can generate the correlations between money growth and in-flation that we stress. However, in contrast to in the Japanese data, money growth does not Granger cause inflation in Woodfords' model (see the discussion in the next section).
20 We employ the notion of causality introduced by Granger (1969, 1980). Money growth is said to cause inflation if it contains information about future inflation that is not contained in some past values of π. The extent and direction of cau-sality can differ between frequency bands (Geweke 1982, Granger and Lin, 1995). The fact that a stationary series is effectively the sum of uncorrelated components, each of which is associated with a single frequency ordinate, al-lows the full causal relationship to be decomposed by frequency.32
Since the causal relation between money growth and inflation could be influ-enced by third variables, we follow Geweke (1984) and investigate Granger causality across frequencies in a vector autoregression (VAR) containing infla-tion, money growth, the output gap and the change in the logarithm of the inter-est rate, i.e. we condition on the output gap and the change in the log interinter-est rate when measuring causality between πt and μt.33
The frequency-wise measure of causality suggested by Geweke (1982) and Hosoya (1991) is defined as:
(10)
where Ψ12 and Ψ11 are obtained from the moving average representation of a VAR that includes the variables of interest (in our case πt and μt) together with the conditioning variables, Zt,
(11)
32 Though the component of a series in a certain frequency band cannot be estimated without the use of a two-sided filter which destroys the chronological aspect of the causal definition, it is possible to deduce causal relationships at different frequencies without estimation of the series' components, as it is done in the band spectrum regressions.
33 We do not condition on output growth and the output gap simultaneously since these va-riables are both derived from real GDP. Furthermore, since output growth is obtained by first-differencing the (logarithm of) output, most of the power at low frequencies is removed so the variable is uninformative about the direction of causality at business-cycle and lower frequencies.
21 where the Ψij(L), i, j = 1, 2, 3 are polynomials in the lag operator, L, and η1, η2, η3 are the orthogonalized shocks.34
Money growth Granger-causes inflation if Ψ12(L) is non-zero. Breitung and Candelon (2006) show that the hypothesis Mμ→π(ω)=0 is equivalent to a linear restriction on the VAR coefficients and that its significance can be tested by a conventional Wald test. To assess the significance of the causal relationship we compare the causality measure for ω∈(0,π) with the critical value of a χ2 -distribution with two degrees of freedom, which is 5.99.
Figure 5 shows the causality measure over frequencies from zero to π. The Akaike criterion indicates a lag length of five for the VAR underlying the causal-ity test. We find significant Granger causalcausal-ity from money growth to inflation at low frequencies, and again at frequencies above 0.6π, which corresponds to three quarters, whereas between 13 and 3 quarters no significant Granger cau-sality is found.35 By contrast, there is no Granger causality from inflation to money growth at any frequency. We also test Granger causality from the output gap to inflation and conversely. The causal relationship from the output gap to inflation is significant except for frequencies between ten and five quarters and shows a peak at the business cycle frequency of 20 quarters. We thus find that output gaps predict inflation at higher frequencies than money growth.
7. Conclusions
The empirical work presented in this paper can be summarised as follows. First and most importantly, the band spectral regressions indicate that money growth is correlated – and output growth is inversely correlated – with inflation in the low frequency band, in particular when that is defined as frequencies of four years or more. Furthermore, that correlation reflects unidirectional Granger cau-sality from money growth to inflation, implying that money growth does contain information about future inflation that is not already embedded in inflation. In the
34 That is, the VAR reduced-form errors are transformed into the orthogonalized errors by mul-tiplying them with the lower triangular matrix from a Choleski decomposition of the reduced-form covariance matrix.
35 Measuring frequency, ω, in fractions of π , periodicity in quarters is given by 2π/ω. Thus, a frequency of ω=0.1π corresponds to a periodicity of 20 quarters.
22 high frequency band the quantity-theoretic variables appear to be of little signi-ficance for inflation.
Second, and focussing on the case in which the distinction between high and low frequency is drawn at four years, the output gap is highly significant at high frequencies but not at low frequencies. Moreover, there is unidirectional Gran-ger causality from the output gap to inflation.
Third, in modelling headline inflation it seems useful to focus on the low-frequency components of money and output growth, and on the output gap. The results indicate that in the long run, a one percentage point increase in the growth rate of money at low frequencies leads to an equal increase in headline inflation. An increase in the low-frequency component of output growth de-presses inflation proportionally, as suggested by the quantity theory.
Fourth, the opportunity cost of holding M2+CD in Japan is better modelled by assuming that it is the logarithm of the interest rate, as opposed to the interest rate itself, that matters. The importance of this specification is clearest once da-ta for the recent period of quantida-tative easing are considered.
While these results are supportive of the notion that money growth does contain information useful for predicting future inflation, since they say nothing about whether money growth is controllable, we emphasise that they do not imply that it would be desirable for the BOJ to target money growth. Nor do they imply that money growth should have any “special” status in the conduct of policy; like other macroeconomic variables, the weight attached to it in the policy process should presumably depend on how closely it is related to inflation.
Finally, we note that we have not explored whether the relationship between money growth and inflation is stable over time. The main reason for this is that since we focus on the low-frequency relationship between the variables, there is in fact little information about that relationship and it is consequently difficult to test whether it has shifted over time. Of course, this is merely the econometric equivalent of the fact that, since the relationship between money growth and inflation involves “long and variable lags”, it is difficult to know in real time whether and, if so, how the relationship has changed over time.
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28 Tables and Figures
Table 1. Unit root tests Sample period: 1970Q1 to 2005Q4.
ADF PP ERS KPSS SIC lag
De-term.
Inflation -2.60 -6.40* -2.36* 1.67* 4 C
Money growth -2.04 -2.04 -0.98 9.94* 0 C
Output growth -3.97* -9.29* -3.40* 1.26* 2 C
Output gap -5.21* -3.69* -3.66* 0.05 3 C
Log interest rate 1.60 0.40 2.18 1.73* 5 C
Change in log interest
rate -8.25* -9.85* -8.14* 0.25 4
C
Real money -1.41 -2.01 -1.06 0.38* 5 T
Real money growth -5.00* -6.22* -3.28* 0.37 4 C
Velocity -3.26 -2.08 -3.04* 0.25* 2 T
Velocity change -5.37* -8.59* -3.15* 0.07 6 C
Note: The last column indicates the number of lags included in the test, which were chosen by the Schwarz information criterion (SIC). The last column indicates whether a constant (C) or a trend and a constant (T) are included in the test regression. The 5% critical values for the tests including a constant (a trend) are -2.89 (-3.45) for the Augmented Dickey-Fuller (ADF) and the Phillips-Perron (PP) test, -1.95 (-2.89) for the Elliot, Stock and Rotenberg (ERS) test and 0.46 (0.15) for the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test. An asterisk, “*”, indicates the rejection of the null hypothesis at the 5% level.
29 Table 2. Band spectrum regressions
Estimates of πti =αμiμti−1+αγiγti−1+αggti−1+εti, i = LF, HF Sample period: 1970Q2 to 2005Q4.
Low frequency: 2 to ∞ years High frequency: 0.5 to 2 years Output growth -0.99**
(0.31) Output growth -1.21**
(0.56)
Note: The dependent variable is the inflation rate at the respective frequency band. All variables are deviations from their sample means. A * indicates significance at the 5%, ** significance at
30
the 1% level. For the regression using adjusted money growth we correct standard errors for the uncertainty related to the first-step estimate of the interest elasticity of money demand (see Murphy and Topel 1985).
Table 3. Two-pillar Phillips curve
Estimates of t
Sample period: 1970Q2 to 2005Q4.
Split at the 2-year frequency Split at the 4-year frequency
Note: The dependent variable is the headline inflation rate. All variables are deviations from their sample mean. Standard errors in parentheses; * indicates significance at the 5%, ** signifi-cance at the 1% level. The last row shows the p-value from a test that the long-run coefficients of money growth and output growth are unity and minus unity, respectively. For the single-equation estimation we use a F-test, for the system estimation a likelihood ratio test. For the regression using adjusted money growth we correct standard errors for the uncertainty related to the first-step estimate of the interest elasticity of money demand (see Murphy and Topel 1985).
31 Figure 1. Data
Inflation
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.016
Interest rate (solid line) and log interest rate (dashed line)
1970 1974 1978 1982 1986 1990 1994 1998 2002 0.00
Output gap: Spectral filter (solid line) and HP (dashed line)
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.03
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.01
Interest rate change (solid line) and change of log interest rate (dashed line)
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.03
Centered 5q M A output growth (solid line) and output growth (dashed line)
1970 1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 2003 -0.036
Figure 2. Inflation and money growth: low (left panel) and high frequency (right panel)
Inflation (solid line) and money growth (dashed line)
1970 1975 1980 1985 1990 1995 2000 2005 -0.018
-0.02 -0.01 0.00 0.01 0.02
-0.015
32 Figure 3. Inflation and output gap: low (left panel) and high frequency
(right panel)
Inflation (solid line) and output gap (dashed line)
1970 1975 1980 1985 1990 1995 2000 2005 -0.015
Figure 4. Interest elasticity of money demand
Recursive coefficient on the interest rate
1995 1997 1999 2001 2003 2005
-4
Recursive coefficient on the log interest rate
1995 1997 1999 2001 2003 2005
-25
Note: The figure shows recursive estimates of the interest elasticity of money demand with their 95% confidence bounds, estimated by DOLS with two lags and leads of the differenced regres-sors and an AR(4) correction for the error term.
33 Figure 5. Causality
Money growth and inflation
0 0.2π 0.4π 0.6π 0.8π π
0 2 4 6 8
10 Output gap and inflation
0 0.2π 0.4π 0.6π 0.8π π
0.0 2.5 5.0 7.5 10.0 12.5 15.0
Note: The solid line shows the causality measure from money growth and the output gap to inflation, the dashed line the causality measure from inflation to the respective variable. The causality measures are derived from a VAR including inflation, money growth, the output gap and the change in the logarithm of the interest rate. The horizontal line represents the 5% criti-cal value. The horizontal axis shows the frequency ordinates as fractions of π.